Weighted $$L^{p}$$ estimates on the infinite rooted k-ary tree

Abstract

In this paper sufficient conditions for weighted weak and strong type (p, p) estimates with \(p>1\) for the centered maximal function on the infinite rooted k-ary tree are provided. Here the line initated by the authors and Safe in Ombrosi et al. (Int Math Res Not IMRN 4:2736–2762, 2021) and providing a further extension of the use of techniques due to Naor and Tao in (J Funct Anal 259(3):731–779, 2010) is continued. The fact that the class of weights from the sufficient conditions is wider for \(L^p\) estimates than the one obtained in [16] is established as well. Some results highlighting the pathological nature of the weighted \(L^p\) theory in this setting are settled. It is shown that the \(A_p\) condition is no precise in this setting, since there exist weights such that the \(L^p\) boundedness holds but the \(A_p\) condition is not satisfied. It is also shown that the Sawyer type testing condition is not sufficient either for the strong type to hold and also that strong and weak type estimates are not equivalent in this setting. It will be shown as well that the one weight results can be extended to the two weight setting.